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Statistical analysis of response to fire cues
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Fire Safety Journal 39 (2004) 663–688 www.elsevier.com/locate/firesaf
Statistical analysis of response to fire cues A.M. Hasofer, D. Bruck CESARE, Victoria University, P.O. Box 14428, MCMC Melbourne, Victoria 8001, Australia Received 13 May 2003; received in revised form 3 May 2004; accepted 9 June 2004 Available online 25 August 2004
Abstract A statistical analysis of data on recognition of fire cues during sleep was carried out to determine the influence of the type of cue, sex and age on response. Four cues were used: crackling, shuffling, a flickering light and smell. The analysis was carried out on the observed data as well as on the corresponding parameters of a stochastic model of response previously developed. Linear models and, where required, a generalized linear model were used. At the low intensity levels used the most effective cues were auditory (crackle, followed by shuffle) while the two least effective were light and smell. Females had a waking up probability consistently higher than males, as well as a shorter response time. The influence of age was borderline for all cues. An important practical conclusion of the study is that a low level flickering light and fire smells are unlikely on their own to arouse sleeping people. r 2004 Elsevier Ltd. All rights reserved. Keywords: Fire cues recognition; Waking up probability; Response time; Influence of sex and age; Linear models; Stochastic modelling
1. Introduction In an earlier paper [1] a stochastic model for the time to awaken in response to a fire alarm was developed, based on a model originally developed by Ratcliff and Murdock in 1976 for analysing two-choice decisions [2]. The model used is called a memory retrieval theory because it considers an item recognition task as testing a Corresponding author. Tel.: +61-3-9216-8033; fax: +61-3-9216-8058.
E-mail address: [email protected] (A.M. Hasofer). 0379-7112/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2004.06.002
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single probe item against a group of items in memory which has been designated as the memory search set. The retrieval process consists of comparing a collection of features of each element of the memory search set with the probe item. It is mathematically identical to the classical ‘‘gambler’s ruin problem’’. The comparison process is a feature-matching process in which probe and memory-set item features are matched one by one. The model was fitted to two sets of data from Bruck and her colleagues [3,4] describing the response of sleeping subjects to a smoke detector alarm. It was validated by visual and statistical means. The model can be used to predict high quantiles of the awakening time, an essential component of the total time to escape, needed to evaluate the risk of death in a compartment fire. In this paper a statistical analysis of data on recognition of fire cues during sleep presented by Bruck and Brennan [5] is carried out. One set of sleeping subjects was exposed to two types of auditory cues: a crackling noise and a shuffling noise, and a visual cue: a flickering light. A second set of sleeping subjects was exposed to an olfactory cue. For each subject, sex and age was recorded. The statistical analysis focusses on modelling the effect of three factors: cue, sex and age on: (1) the observed data: whether the subjects wake up or not, and if they wake up, the mean response time, (2) the model parameters derived from the observed data: (a) the so-called ‘‘matching’’ probability, (b) the wake up threshold, (c) the length of a step in the ‘‘matching’’ process, (3) two derived measures (which are conditional on waking up): the mean number of steps to reach the threshold, and the ninety ninth percentile of the response time. Linear models and, where required, a generalized linear model are used. Analysis of variance (and analysis of deviance for the generalized linear model) are used to determine the statistical significance of the factor effects. Graphical representation of the fitted values provides valuable insights into the factor effects. The relation between the observations, the model parameters and the derived measures is interpreted. Conclusions are drawn and suggestions for further work put forward.
2. Description of the data 2.1. Experiment 1: Responsiveness to auditory and visual cues Participants: Thirty three adult volunteers aged 25–55 years (mean age 43.2 years) who self-reported normal hearing and normal sleep patterns were tested in their homes. There were 15 males and 18 females. Cues: Two auditory cues, a crackling noise akin to the early stage of a timber fire and a ‘shuffling’ noise, as well as a flickering light cue were used. The sounds were
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chosen to represent sounds from a fire. The first is subsequently referred to as ‘‘crackling’’ and the second as ‘‘shuffling’’. The sounds were edited to 30 s from a sound effects recording for movies and videos, with the selection of sections with less variation. The crackling sound varied at the pillow across the 42–48 dBA range with the highest sound being 58 dBA at the 25th second. This sound was sharper and more varied than the shuffling sound, which varied from 43–45 dBA with a few higher points of 52 dBA. The flickering light was directed towards the ceiling and reached the pillow at 5 lux or less. There was considerable variation in the strength of this cue, due to variation in bedroom design and reflectiveness of the ceiling. In some situation the light reading from the pillow was as low as 1 lux. (one lux is approximately the light of one candle flame at a distance of one meter.) Procedure: Cues were presented in random order on three nights within a five night period. Cue delivery occurred five times over a five-minute period lasting for 30 s each time with 30 s between presentations. Participants activated the program (unaware of whether it was delivering a signal that night or not) each night by pressing a ‘sleep’ button on the monitor when they were preparing to go to sleep. Cue presentation began 240 min after this time. This four-hour time period was chosen to increase the likelihood of participants being in Stage 2 or REM sleep when the cues were delivered. These two stages have similar arousal levels [6]. If awakenings were timed to occur earlier in the night then higher arousal thresholds would be expected as deeper sleep would be more likely. (In a normal night’s sleep the deeper Stage 4 sleep is more likely to occur in the first third of sleeping while Stage 2 and REM occur in the last two-thirds.) Participants were told that when they were asleep a cue might be presented that they could see, hear or smell. There would be some nights without cues, though the number could not be stated. Nor could they be told when the signal might occur. If they noticed anything like a signal, they were to press the button by their bed three times. This would stop the delivery and they could go back to sleep. There would be no further cues that night. 2.2. Experiment 2: Responsiveness to an olfactory cue Participants: Twenty young adults, aged 18–26 years (mean age 21.3 years), selfreporting a normal sense of smell and the ability to sleep during the day were involved. There were ten females and ten males. The participants were students from the University. All subjects were asked to restrict their sleep to around 6 h (e.g. from 1 am to 7 am) on the previous night to enhance the likelihood of going to sleep in the afternoon. During preparation for the sleep recordings the olfactory sense of smell of all subjects was somewhat informally checked by determining whether they could smell the alcohol on a cleaning swab. One person could not (due to a cold) and she was eliminated from the study. Cue: The cue presented was the somewhat unpleasant smell of a chemical associated with a smoke taste in food flavourings (Guaiacol). This was prepared in an ethanol base at a 5% concentration level in an aerosol container. A dispenser
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emitted puffs of the mixture at the rate of nine puffs per minute and this odour was dispersed into the bedroom via a duct and fan. Low levels were used, where the parts per million were an average of 0.9 ð0:4Þ at one minute to 6.0 pm ð1:0Þ at 10 min at the pillow. As the human nose is very sensitive, the smell was clearly evident at the pillow 20 s after dispensing began when the concentrations were well below 1 pm. Procedure: Participants attended the sleep laboratory for an ‘afternoon nap’. Sleep stages were recorded via the standard electrode montage on the scalp and face. Subjects began their nap at between 1.30 and 2.15 pm, odour presentations began 90 s after the person entered Stage 2 sleep and continued for a maximum of 10 min. Subjects were asked to press a button three times if they became aware of the presence of a stimulus that they could see, smell or hear. The study provided information about both the time of EEG wakefulness and the behavioural response (button press). Interestingly, three subjects clearly entered EEG wakefulness during the smell presentation but did not press the button. Questioning revealed that they did not believe they had woken up. Accordingly, since this study is essentially behavioural, the three subjects were classified as not waking up.
3. Analysis background The analysis presented in this paper concentrates on the response to single cues. Accordingly, the responses of the 33 subjects in experiment 1 to the three different cues (auditory and visual) were analyzed as if they had been responses from 99 different subjects. For each subject, the age and sex was recorded. The number of subjects responding to all the cues (including the olfactory cue of Experiment 2), after gaps in the data were removed, is shown in Table 1. The total number was 104.
4. Statistical tools for analysis of waking probability To analyse the waking probability it is necessary to investigate the dependence of a variable that takes just two values: waking and not waking, on a continuous variable: age, as well as on various levels of two categorical factors: sex and cue. The
Table 1 Number of subjects by sex and cue Cue
Male
Female
Light Crackle Shuffle Smell
13 12 12 10
15 17 15 10
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two values of the response are coded as 0 (not waking) and 1 (waking). This type of dependent variable is known as a binomial variable. To analyse such a set of data, the classical linear model is not suitable, because it presupposes that the dependent variable is normally distributed for each combination of levels and that its variance is constant. The appropriate model to carry out such an investigation is known as the generalized linear model. The required theoretical foundation was developed by Nelder and Wederburn [7]. For a full exposition of the generalized linear model see [8]. The difference between the linear model and the generalized linear model is as follows: The linear model provides a way of estimating a response variable,Y, conditional on a linear function of the values, x1 ; x2 ; . . . ; xp , of some set of predictors variables. The variance of Y is assumed constant. The generalized linear model provides a way to estimate a function, gð Þ, (called the link function) of the mean response m as a linear function of the values of some set of predictors. The variance of Y is a function V ð Þ of the mean response m. In this paper, the logistic model, for which the link function (the so-called logit function) is given by gðmÞ ¼ logðm=ð1 mÞÞ and the variance function is V ðmÞ ¼ mð1 mÞ. It is used to deal with a binomial response. In the generalized linear model, the sum of squares attributed to each underlying factor is replaced by a deviance term that has a similar interpretation. It is however important to note that each one of the successive deviance terms represents the additional variability that is attributable to the considered factor, once the variability attributable to the previous factors has been accounted for. On account of this, the size of the various deviance components depends somewhat on the order of the terms in the regression equation, which must be chosen judiciously. The significance of each factor is tested by means of a chi squared test. The generalized linear model allows for the study of the interaction between factors. However, because of the comparatively small amount of data, no interactions were included. The computer program used to implement the analyses was S-PLUS Version 6.
5. Results for the waking probabilities The binomial response variable was regressed on two categorical factors: cue and sex, and on age as a continuous variable. The regression on age was carried out as a polynomial regression. After some experimentation it turned out that a polynomial of order three would be suitable. The most significant factor was the cue and the second sex. The analysis of deviance is shown in Table 2. In the rest of the paper, the estimated waking probability will be denoted by pZ . It can be seen that while the cue and sex factors are significant at the 2% level, age is not significant, even at the standard 5% level, although there is clearly some effect. The effect of cue, sex and age are illustrated in Figs. 1 and 2 where the fitted values of the response probability are plotted against cue and age for males and females,
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Table 2 Analysis of deviance of waking probability
NULL Cue Sex Age
Degrees of freedom
Deviance
Residual deviance
3 1 3
20.50 5.66 4.34
134.17 113.66 108.00 103.66
Probability of chi squared
0.0001 0.017 0.23
WAKING PROBABILITIES FOR MALES BY AGE AND CUE 1.00
CRACKLE
Probability of waking
0.80
0.60 SHUFFLE
0.40
0.20
SMELL
LIGHT
0.00 20
30
40
50
Age Fig. 1. Fitted waking probabilities of male subjects by age and cue.
respectively. Continuous curves are obtained by using cubic spline interpolation to derive intermediate values between the observations. A larger sample would have provided more reliable results for the effect of age. With the data at hand, any conclusions about the effect of age should be taken as preliminary only. In order to tabulate the waking probabilities by cue, sex and age, subjects were grouped by age as shown in Table 3. Table 4 gives the fitted values of the waking probabilities classified by age group, sex and cue. The table shows that the mean female waking probability is consistently larger than the mean male waking probability for each cue and each age group,
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669
WAKING PROBABILITIES FOR FEMALES BY AGE AND CUE CRACKLE
0.80 Probability of waking
SHUFFLE
0.60 SMELL
0.40
LIGHT
0.20
0.00 20
30
40
50
Age Fig. 2. Fitted waking probabilities of female subjects by age and cue.
Table 3 Age groups and number of subjects in each age group Age group
Number
p30 31–42 43–46 47–50 51þ
30 23 19 21 11
Table 4 Fitted waking probabilities by age group, sex and cue Cue
Light Crackle Shuffle Smell
Sex
Male Female Male Female Male Female Male Female
Age Group p30
31–42
43–46
47–50
51þ
All ages
0.267 0.522 0.848 0.944 0.690 0.870 0.317 0.583
0.170 0.381 0.758 0.904 0.556 0.790 — —
0.224 0.464 0.816 0.930 0.638 0.841 — —
0.356 0.624 0.894 0.962 0.772 0.910 — —
0.546 0.783 0.949 0.982 0.880 0.957 — —
0.288 0.551 0.836 0.939 0.673 0.862 0.317 0.583
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Table 5 Analysis of variance of response time
Cue Sex Age Residuals
Degrees of freedom
Sum of squares
Mean squares
F value
Pr(F)
3 1 3 60
117,793 2,659 22,850 177,639
39,264 2,659 7,617 2,961
13.3 0.90 2.57
0.000001 0.35 0.06
6. Results for the mean response time Once the subjects who did not wake up were eliminated, there remained a sample of 68 subjects for whom a response time (in seconds) had been recorded. The analysis was carried out using a standard linear model: the response time was regressed as before on two categorical factors: cue and sex, and on age as a continuous variable (represented as in the previous section by a polynomial of order 3). The analysis of variance is given in Table 5. Here the conventional sum of squares analysis is carried out, and significance is tested by means of an F test. It can be seen that the type of cue remains the most significant factor. However, it is now age that takes the second place and just misses becoming significant, while sex is no longer significant. The effect of cue and age is illustrated in Fig. 3 where the fitted values of the mean response time are plotted against cue and age. Continuous curves are obtained by using cubic spline interpolation to derive intermediate values between the observations. A larger sample would have provided more reliable results for the effect of age. With the data at hand, any conclusions about the effect of age and sex should be taken as preliminary only. Table 6 gives the fitted values of the mean response time classified by age group, sex and cue, using the same age groups as for the waking probabilities. The table shows that the mean female response time is consistently shorter than the mean male response time for each cue and each age group, even though here this difference does not satisfy the criteria of statistical significance.
7. The stochastic model for the response and its parameters One aim of the research reported in this paper is to examine the dependence of the parameters of the stochastic model for the response to fire alarms proposed in [1] on the three considered factors: cue, sex and age. This will provide valuable insight into the underlying determinants of the response variability. As mentioned in Introduction, the model used in this paper is called a memory retrieval theory because it considers an item recognition task as testing a single probe item against a group of items in memory which has been designated as the memory
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COMBINED MEAN RESPONSE TIME BY AGE AND CUE SMELL
Mean response time (secs)
150
LIGHT 100
50
CRACKLE
SHUFFLE 0 20
30
40
50
Age Fig. 3. Combined fitted mean response time by age and cue.
Table 6 Fitted mean response time by age group, sex and cue Cue
Light Crackle Shuffle Smell
Sex
Male Female Male Female Male Female Male Female
Age Group p30
31–42
43–46
47–50
51þ
All ages
— 116.86 71.77 57.98 — 52.28 150.52 136.74
93.26 — 34.38 20.59 28.68 14.89 — —
84.58 70.79 25.70 11.91 20.00 6.21 — —
85.57 71.78 26.69 12.90 20.99 7.20 — —
119.38 105.59 60.50 46.71 54.80 41.01 — —
98.57 85.59 38.14 25.16 32.04 19.05 150.52 136.74
search set. The retrieval process consists of comparing a collection of features of each element of the memory search set with the probe item. In the application of the model to the response of subjects to a fire cue the probe item is identified with the stimulus induced in the sleeping subject by the cue. This stimulus is compared with the items of the memory search set. Comparison of a probe to a memory-set item proceeds by the gradual accumulation of evidence, that is, information representing the goodness of match, over time. It is easiest to conceptualize the comparison process as a feature-matching
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process in which probe and memory-set item features are matched one by one. A count is kept of the combined sum of the number of feature matches and nonmatches, so that for a feature match, a counter is incremented, and for a feature nonmatch, the counter is decremented. The counter begins at some starting value Z, and if a total of A counts are reached, the probe is declared to match the memory-set item (A Z more feature matches than nonmatches). But if a total of zero counts are reached, an item nonmatch is declared. The response is thus modelled by a random walk bounded from above and below. The lower boundary is taken to be 0. The upper boundary is denoted by A and the starting value by Z. The process ends when either of the boundaries is reached. If the lower boundary is reached first the subject dismisses the stimulus and continues to sleep. If, on the other hand, the upper boundary is reached first the subject wakes up. Thus, the upper boundary A can be thought of as a threshold of sensitivity to the stimulus. The random walk is described by four parameters: (1) Z, the starting value, (2) A, the wake up threshold, (3) the probability p of a match between the stimulus and an item of the memory set, resulting in an upward jump of one unit at each step. A non-match, that occurs with probability 1 p, results in a downward jump of one unit. (4) the length d of one step. It was decided, after some preliminary analysis, to fix the starting value Z to an appropriate value. The three remaining parameters can then be estimated from the knowledge of: (1) the probability pZ of the subjects responding to the alarm, (2) the mean and standard deviation of the response time, for subjects who have responded.
8. The variance of the response time So far in the paper, the probability of response and the mean response time have been determined for each category of subjects. But because of the comparatively small sample available, it is not possible to estimate the standard deviation of the response time separately for each category with any confidence. However, in view of the fact that the cue factor is highly significant for the response time, the reasonable path is to estimate the standard deviation for each cue, using the sample of all subjects receiving the cue. It is estimated as the standard deviation of the difference between the observed response time and the fitted mean. The estimated standard deviations are given in Table 7. The overall standard deviation is 51.49 s.
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Table 7 Estimated standard deviation for each cue Cue
Standard deviation
Light Crackle Shuffle Smell
92.71 28.23 22.86 81.18
9. Results for A, p and d The value of Z was chosen to be 50. This was a compromise between two conflicting considerations. On the one hand, choosing Z too small would make the random walk too coarse. On the other hand, making Z too large would vastly increase the amount of computation required without achieving a more significant fit. The choice was also in line with the results of the analysis in [1], where A was chosen to be 100 and the two values of Z evaluated were 75 and 72. Using the fitted response probability, the fitted mean, and the estimated standard deviation, values of A, p and d (in seconds) were calculated for each subject who woke up and each cue, using the method described in [1]. Dependence of A, p and d on cue, sex and age was analysed, using a standard linear model, with cue and sex being treated as categorical factors and age as a continuous variable, modelled as previously with a polynomial of order three. It would appear at first sight that the analysis on p should be carried out using the logistic model, as p is a probability. However, closer analysis of p reveals that its values are clustered around the value 0.5. Indeed, 57 out of 68 values of p lie in the interval (0.45,0.55). The histogram of these 57 values is given by in Fig. 4. But it is well-known that the logistic transformation is practically linear when the dependent variable is in the neighbourhood of 0.5, so that using it does not improve the fit over the standard linear model. On the other hand, the standard linear model is more robust than the generalized linear model because it makes fewer distributional assumptions. It was therefore decided to use the standard linear model for p as well. The analysis of variance for A, d and p is given in Tables 8–10. The analysis of variance for the threshold A shows that age is by far the most significant factor, followed by cue. Sex just misses out to be significant at the 5% level. The effect of cue, sex and age on A are illustrated in Figs. 5 and 6. Since the effect of sex turned out to be barely significant, the effect of cue and age on A for the combined sample of males and females is also illustrated in Fig. 7. As in the previous figures, continuous curves are obtained by using cubic spline interpolation. Table 11 gives the fitted values of A classified by age group, sex and cue. The table shows that the mean female A is consistently lower than the mean male A for each cue and each age group. A larger sample would probably have yielded a significant difference.
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HISTOGRAM OF p FOR 0.45
25
20
15
10
5
0 0.45
0.47
0.49
0.51
0.53
0.55
p
Fig. 4. Histogram of p for 0:45opo0:55.
Table 8 Analysis of variance of A
Cue Sex Age Residuals
Degrees of freedom
Sum of squares
Mean squares
F value
Pr(F)
3 1 3 60
35,337 4,629 54,778 71,056
11,779 4629 18,259 1,184
9.95 3.9 15.4
0.00002 0.053 0.00
Table 9 Analysis of variance of d
Cue Sex Age Residuals
Degrees of freedom
Sum of squares
Mean squares
F value
Pr(F)
3 1 3 60
130.1 0.116 47.46 738.4
43.4 0.116 15.82 12.31
3.52 0.009 1.28
0.020 0.92 0.29
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Table 10 Analysis of variance of p
Cue Sex Age Residuals
Degrees of freedom
Sum of squares
Mean squares
F value
Pr(F)
3 1 3 60
0.069 0.0028 0.0083 0.40
0.023 0.0028 0.0028 0.0067
3.48 0.43 0.42
0.02 0.52 0.74
FITTED A FOR MALES BY AGE AND CUE CRACKLE
150 SMELL
Fitted.A
SHUFFLE
100 LIGHT
50
0 20
30
40
50
Age Fig. 5. Fitted A for male subjects.
The main conclusions from the analysis of the dependence of the threshold A on cue and age are as follows: (1) The order of the cues in descending values of fitted A is shuffle, crackle, light and smell. (2) The lowest values of A for all cues (except for smell, for which the range of ages does not extend beyond 26) are in the early forties, with A increasing with younger as well as with older age. It is possible that the higher threshold for younger people can be attributed to an increased delta power (see [9]). This means that their sleep is deeper and consequently a larger net number of matches is required for response. On the other hand, the higher threshold for older people
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FITTED A FOR FEMALES BY AGE AND CUE
150
Fitted.A
SHUFFLE 100
SMELL CRACKLE
LIGHT
50
0 20
30
40
50
Age Fig. 6. Fitted A for female subjects.
COMBINED A BY AGE AND CUE
150
Fitted.A
SMELL
CRACKLE
SHUFFLE
100
LIGHT
50
0
20
30
40 Age
Fig. 7. Combined fitted A.
50
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Table 11 Values of fitted A by age group, sex and cue Cue
Sex
Light Crackle Shuffle Smell
Age Group
Male Female Male Female Male Female Male Female
p30
31–42
43–46
47–50
51þ
— 132.74 179.83 157.92 — 174.43 166.50 144.59
75.46 — 100.63 78.72 117.15 95.24 — —
58.95 37.04 84.13 62.22 100.64 78.73 — —
65.79 43.88 90.96 69.05 107.48 85.57 — —
111.76 89.85 136.94 115.03 153.45 131.54 — —
Table 12 Mean values of p
p Meanj0:5 pj
Light
Crackle
Shuffle
Smell
0.46 0.0417
0.54 0.0430
0.48 0.05
0.50 0
Table 13 Mean values of d
Mean d
Light
Crackle
Shuffle
Smell
0.21
0.14
0.06
0.030
might be attributed to a requirement for stronger evidence before they decide to respond. In the analysis of variance of p and d, it turned out that only cue was significant, while sex and age were not. Two values of d, 26.02 and 16.50,were considered to be outliers. Their appearance is probably due to the very rough estimation of the standard deviation as well as the fact that the values of A, p and d were calculated from a discrete approximation that could be misleading for certain values of the input. After their removal the 66 remaining values of d lay in the interval (0.007,1.18). The mean values of p and d for each cue are shown in Tables 12 and 13. The order of the cues in descending values of mean p is: crackle, smell, shuffle, light.
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COMBINED FITTED p BY AGE AND CUE
CRACKLE
Fitted.p
0.55
0.50
SMELL SHUFFLE
0.45
LIGHT
0.40 20
30
40
50
Age Fig. 8. Combined fitted p.
Although the dependence of p on age is not statistically significant, it is nevertheless of interest to look at the graph of fitted p versus cue and age. It is given by Fig. 8. There is also some difference between the mean values of p for males and females. They are 0.494 and 0.506, respectively. The order of cues in descending values of mean d is: light, crackle, shuffle, smell. What is particularly noticeable, (and perhaps somewhat puzzling) is the inordinately high value of d for light, indicating that, in the context of waking up to a light cue, a visual stimulus is processed far more slowly than an auditory or olfactory stimulus.
10. Results for the number of steps to response While the probability of response depends only on the two underlying parameters A and p, the response time depends on all three parameters A, p and d. This makes it difficult to interpret the dependence of the response time on cue, sex and age in terms of the dependence of the underlying parameters. One way of simplifying the interpretation is to evaluate the number of steps to response, which will be denoted by N. This is easily accomplished by dividing the response time for each subject by the evaluated length of step d. Analysis of the dependence of N on cue, sex and age was carried out, using a standard linear model with cue and sex treated as categorical factors and age as a
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Table 14 Analysis of variance of N
Cue Sex Age Residuals
Degrees of freedom
Sum of squares
Mean squares
F value
Pr(F)
3 1 3 60
225,605,699 37,585,271 62,758,913 290,354,873
75,201,900 37,585,271 20,919,638 4,839,248
15.5 7.8 4.3
0.0000001 0.007 0.008
FITTED N FOR MALES BY AGE AND CUE SMELL
Fitted.N
6,000
4,000
SHUFFLE CRACKLE
2,000
LIGHT
0
20
30
40
50
Age Fig. 9. Fitted N for male subjects.
continuous variable modelled as previously with a polynomial of order three. The analysis of variance is given in Table 14. Here it turns out that all three factors are highly significant. The effect of cue, sex and age on N is illustrated in Figs. 9 and 10.
11. The ninety ninth percentile of the distribution of response time It was pointed out in [1] that one of the most important aims of the stochastic analysis of the response time to fire cues was to estimate the high quantiles of the distribution of the response time, since it is those who respond the most slowly who are the most at risk.
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FITTED N FOR FEMALES BY AGE AND CUE
SMELL
Fitted.N
6000.00
4000.00
2000.00 SHUFFLE CRACKLE 0.00
LIGHT 20
30
40 Age
50
Fig. 10. Fitted N for female subjects.
Following [1], the ninety ninth percentile of the distribution, denoted by q99 , was taken as a representative of the upper tail of the distribution. It was evaluated for all subjects who woke up, using the values of A, p and d previously obtained and the formulae in [1]. There were eight subjects where the value of p was so low that the value of q99 was essentially infinite. They were excluded from the sample, leaving 60 subjects for analysis. In a risk analysis of response, those eight subjects should be classified as not waking up. Analysis of the dependence of q99 on cue, sex and age was carried out, using a standard linear model with cue and sex treated as categorical factors and age as a continuous variable modelled as previously with a polynomial of order three. The analysis of variance is given in Table 15. Here it turns out that while cue and sex are significant, age is not. This is consistent with the results of Table 2, which gives the analysis of deviance of the waking probability, since the subjects who do not wake up may be thought of as subjects with an infinite response time. The effect of cue, sex and age on q99 is illustrated in Figs. 11 and 12. Table 16 gives the fitted values of q99 classified by age group, sex and cue. The table shows that the mean female q99 is consistently lower than the mean male q99 for each cue and each age group.
12. Summary of dependence on cue, sex and age The dependence of the observed variables pZ and T as well as the model parameters A, p and d, the derived variable N and the ninety ninth percentile q99 on the three factors considered, cue, sex and age, is summarized in Table 17.
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Table 15 Analysis of variance of q99
Cue Sex Age Residuals
Degrees of freedom
Sum of squares
Mean squares
F value
Pr(F)
3 1 3 52
1726390 12090 12092 104349
575464 12090 4031 2006.7
287 6.02 2.01
0.00000 0.017 0.124
99TH QUANTILE OF RESPONSE TIME FOR MALES
600
99th quantile (secs)
LIGHT
SMELL 400
CRACKLE
200
SHUFFLE 0 20
30
40
50
Age Fig. 11. Fitted q99 for male subjects.
13. Theoretical dependence of pZ and N on p and A To be able to interpret the relationship between the observed parameters, namely the probability of waking and the mean response time, in terms of the model parameters A, p and d, it is necessary to look at the theoretical relationship between them. Fig. 13 illustrates the dependence of the probability of waking pZ on p and A while Fig. 14 illustrates the dependence of the mean number of steps N on p and A. From Fig. 13 we see that pZ varies directly with p and inversely with A. This is in accordance with common sense. However, there is much difference between the sensitivity of pZ to A and to p. This is illustrated in Figs. 15 and 16, which plot the values of the derivatives of logðpZ Þ with respect to logðAÞ and logðpÞ against p.
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99TH QUANTILE OF RESPONSE TIME FOR FEMALES
99th quantile (secs)
600
SMELL
LIGHT
400
200
CRACKLE
SHUFFLE
0 20
30
40
50
Age Fig. 12. Fitted q99 for female subjects.
Table 16 Values of fitted q99 by age group, sex and cue Cue
Light Crackle Shuffle Smell
Sex
Male Female Male Female Male Female Male Female
Age Group p30
31–42
43–46
47–50
51þ
All ages
— 491.23 182.15 143.01 — 116.23 506.83 467.69
480.27 — 132.04 92.90 105.27 66.12 — —
— 446.27 137.19 — 110.42 71.27 — —
511.43 472.29 163.21 124.07 136.43 97.29 — —
503.28 464.14 155.06 115.91 128.28 89.14 — 467.69
500.91 471.71 147.07 117.88 117.43 88.24 506.83
It must be pointed out that these derivatives measure the relative change of pZ with respect to the relative change of A and p, which allows comparison of sensitivity with respect to the variables by direct comparison of the derivatives. It can be seen that pZ is far less sensitive to relative variations in A than it is to relative variations in p, particularly for p near 0.5. On the other hand, from Fig. 14 we see that while N varies directly with A, it varies inversely with j0:5 pj, i.e., with the distance of p from 0.5. Superficially, one would
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Table 17 Summary of dependence on cue, sex and age Dependence on
Waking probability pZ Mean response time T Threshold A Probability of match p Length of step d Mean number of steps to waking N Ninety ninth percentile q99
Cue
Sex
Age
Strong Strong Strong Strong Strong Strong Strong
Significant Weak Borderline significant Very weak Very weak Strong Strong
Weak Borderline significant Strong Very weak Weak Strong Weak
DEPENDENCE OF pZ ON p AND A 1.0
0.8
pZ
0.6
0.4 A=75
0.2
A=100 A=150
0.0 0.46
0.47
0.48
0.49 p
0.50
0.51
0.52
Fig. 13. Dependence of pZ on p and A.
have thought that N should vary inversely with p monotonically. This, however, ignores the fact that N is the mean response time conditional on waking up. The average is taken only over the sample paths that end up in waking up. For small values of p most subjects do not wake up. The only subjects who wake up are those who at the beginning of the matching process happen to have a large enough proportion of successful matches at the beginning of the process to bring them to the threshold A. This tends to shorten the response time conditional on waking up.
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DEPENDENCE OF N ON p AND A A=150
6000 A=125
N
4000 A=100
2000 A=75
0 0.43
0.45
0.47
0.49
0.51
0.53
0.55
0.57
p Fig. 14. Dependence of N on p and A.
RELATIVE DERIVATIVE OF pZ W.R.T. A
d(log pZ)/d(log A)
0
-10 A=75
-20 A=100
-30 A=150
-40 0.43
0.45
0.47
0.49
0.51
0.53
0.55
p Fig. 15. Dependence of the derivative of logðpZ Þ w.r.t. logðAÞ on p.
0.57
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RELATIVE DERIVATIVE OF pZ W.R.T. p 200 A=150
d(log pZ)/d(log p)
150
100 A=100 50 A=75 0 0.43
0.45
0.47
0.49
0.51
0.53
0.55
0.57
p Fig. 16. Dependence of the derivative of logðpZ Þ w.r.t. logðpÞ on p.
14. Interpretation of the variation of waking probability in terms of A and p As pointed out in Section 13, the waking probability depends only weakly on the threshold A, compared to its dependence on p. Indeed Figs. 1 and 2 look remarkably like Fig. 8 as far as order of cues and age dependence is concerned. So we can conclude that the dependence on cue and on age is governed by the matching probability p. Similarly, the difference in waking probabilities between males and females can be attributed, at least in part, to the slightly higher value of p for females, as compared to males. Here, however, the lower threshold for females compared to males will tend to reinforce the effect of a higher value of p. The dependence of pZ on cue, sex and age in terms of A and p is summarized in Table 18.
15. Interpretation of the variation of mean number of steps to response, N, in terms of A and p Here the interpretation is rather straightforward in the light of the comments in Section 13. The general shape of the dependence of N on cue, sex and age is similar to the shape of the dependence of the threshold A. On the other hand, the larger values of meanj0:5 pj for shuffle and crackle compared to light will tend to shift the
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Table 18 Summary of dependence of pZ on cue, sex and age Dependence of pZ in terms of
Cue Sex Age
A
p
No Yes No
Yes Yes Yes
Table 19 Summary of dependence of N on cue, sex and age Dependence of N in terms of
Cue Sex Age
A
p
Yes Yes Yes
Yes No No
curves downwards, and the small value of meanj0:5 pj for smell tends to shift it higher. The dependence of N on cue, sex and age in terms of A and p is summarized in Table 19.
16. Conclusions The main use of the analysis in this paper in fire engineering is to provide precise data on two quantities: (1) the probability of not waking up, (2) the ninety ninth percentile of the response time for the subjects who have woken up. The latter quantity is particularly important as it determines to what extent fire retardation measures should be taken to allow the subjects who wake up to evacuate the fire area before untenable conditions set in. However, that quantity cannot be evaluated without first evaluating the model parameters. In addition, knowledge of the model parameters is needed to evaluate the full probability distribution of the response time, which is needed if the response time study is extended to multiple cues, as will be proposed in the next section. The analysis in the preceding sections indicates a very significant difference between males and females, both in waking probability and in the response time. Females have a waking up probability consistently higher than that of males in the same categories, which is stastistically significant. Among those who wake up,
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the mean response time of females is consistently shorter than that of males, although the difference is not statistically significant. When it comes to the ninety ninth percentile, however, the response time of females is again consistently shorter than that of males, and the difference is statistically highly significant. In terms of the model parameters, it appears that the greater sensitivity of females is mainly due to a higher value of the matching probability p, although the analysis fails to attribute a statistical significance to the difference. As far as the response time is concerned, the difference is reinforced by the consistently lower value of the threshold for females. The variable of age exerts little influence on the waking probability and an influence on the mean response time of borderline significance. The latter appears to be mainly due to the dependence of the threshold A on age. That threshold is high for the young and the old, reaching a minimum in the early forties. A possible explanation, already given in Section 9, is that it is possible that the higher threshold for younger people can be attributed to an increased delta power. This means that their sleep is deeper and consequently a larger net number of matches is required for response. It is speculated that the higher threshold for older people might be attributed to a requirement for stronger evidence before they decide to respond. The difference between age groups is far less for the ninety ninth percentile of the response time. The most significant difference in the analysis is between the four cues investigated. Indeed, observations as well as model parameters and derived values all show clear differences. It must be pointed out, however, that response to a cue clearly depends on the intensity of the cue. Since the intensity of cue was not varied in the experiment being analysed, it must be kept in mind that all comparisons between cues apply only to the intensities used in the experiment and that these were low level intensities. As far as waking probabilities are concerned, the most effective cues were auditory (crackle, followed by shuffle). The two least effective cues were light and smell. Similarly, for the shortest mean response times as well as the shortest ninety ninth percentile of response time, the cues were again auditory (crackle and shuffle) with smell and light requiring a longer response time. The difference in waking probabilities is clearly attributable to the difference in matching probabilities. However, analysis shows that the threshold for light is actually lower than that of the auditory cues, and the value of meanj0:5 pj is similar for light and the auditory cues. Nevertheless, the response time for light is longer. It turns out that this is due to a much larger value of the step length d for the visual cue. This might be due to the fact that the amount of information contained in visual cues is very different from that contained in auditory or olfactory cues, requiring more time for information processing of the cue. The practical conclusion of the study is that (1) globally speaking the waking probability is unacceptably low for visual and olfactory cues taken singly at the low intensities of this study, making the risk of death unacceptably high,
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(2) from a fire engineering point of view the existence of low level flickering light and fire smells is very unlikely to arouse sleeping people.
17. Proposals for further study The authors propose to conduct in the near future further experiments to evaluate (1) the effect of multiple cues, (2) the effect of varying the intensity of cues, (3) the effect of alcohol. Extensive further experimentation in the area of the processes involved in waking up, coupled with detailed modelling and analysis, is the only way to acquire the knowledge required to develop performance-based guidelines for the design of fire safe dwellings. References [1] Hasofer AM. A stochastic model for the time to awaken in response to a fire alarm. J Fire Protect Eng 2001;11(3):151–60. [2] Ratcliff R, Murdock Jr BB. Retrieval processes in recognition memory. Psychol Rev 1972;83:190–214. [3] Bruck D, Bliss RA. Sleeping children and smoke alarms. Fourth asian-oceania symposium on fire science and technology. Tokyo, May 2000. [4] Bruck D, Horasan M. Non-arousal and non-action of normal sleepers in response to a smoke detector alarm. Fire Safety J 1995;25:125–39. [5] Bruck D, Brennan P. Recognition of fire cues during sleep. In: Shields J, editor. Proceedings of the second international symposium on human behaviour in fire. London: Interscience Communications; 2001. p. 241–52. [6] Zepelin H, McDonald CS, Zammit GK. Effects of age on auditory awakening. J Gerontol 1984;39(3):294–300. [7] Nelder JA, Wedderburn RWM. Generalized linear models. J Roy Stat Soc Ser A 1972;135:370–84. [8] McCullagh P, Nelder JA. Generalized linear models, 2nd ed. London: Chapman and Hall; 1989. [9] A˚stro¨m C, Trojaborg W. Relationship of age to power spectrum analysis of EEG during sleep. J Clin Neurophysiol 1992;9:424–30.
Fire Safety Journal 39 (2004) 663–688 www.elsevier.com/locate/firesaf
Statistical analysis of response to fire cues A.M. Hasofer, D. Bruck CESARE, Victoria University, P.O. Box 14428, MCMC Melbourne, Victoria 8001, Australia Received 13 May 2003; received in revised form 3 May 2004; accepted 9 June 2004 Available online 25 August 2004
Abstract A statistical analysis of data on recognition of fire cues during sleep was carried out to determine the influence of the type of cue, sex and age on response. Four cues were used: crackling, shuffling, a flickering light and smell. The analysis was carried out on the observed data as well as on the corresponding parameters of a stochastic model of response previously developed. Linear models and, where required, a generalized linear model were used. At the low intensity levels used the most effective cues were auditory (crackle, followed by shuffle) while the two least effective were light and smell. Females had a waking up probability consistently higher than males, as well as a shorter response time. The influence of age was borderline for all cues. An important practical conclusion of the study is that a low level flickering light and fire smells are unlikely on their own to arouse sleeping people. r 2004 Elsevier Ltd. All rights reserved. Keywords: Fire cues recognition; Waking up probability; Response time; Influence of sex and age; Linear models; Stochastic modelling
1. Introduction In an earlier paper [1] a stochastic model for the time to awaken in response to a fire alarm was developed, based on a model originally developed by Ratcliff and Murdock in 1976 for analysing two-choice decisions [2]. The model used is called a memory retrieval theory because it considers an item recognition task as testing a Corresponding author. Tel.: +61-3-9216-8033; fax: +61-3-9216-8058.
E-mail address: [email protected] (A.M. Hasofer). 0379-7112/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.firesaf.2004.06.002
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single probe item against a group of items in memory which has been designated as the memory search set. The retrieval process consists of comparing a collection of features of each element of the memory search set with the probe item. It is mathematically identical to the classical ‘‘gambler’s ruin problem’’. The comparison process is a feature-matching process in which probe and memory-set item features are matched one by one. The model was fitted to two sets of data from Bruck and her colleagues [3,4] describing the response of sleeping subjects to a smoke detector alarm. It was validated by visual and statistical means. The model can be used to predict high quantiles of the awakening time, an essential component of the total time to escape, needed to evaluate the risk of death in a compartment fire. In this paper a statistical analysis of data on recognition of fire cues during sleep presented by Bruck and Brennan [5] is carried out. One set of sleeping subjects was exposed to two types of auditory cues: a crackling noise and a shuffling noise, and a visual cue: a flickering light. A second set of sleeping subjects was exposed to an olfactory cue. For each subject, sex and age was recorded. The statistical analysis focusses on modelling the effect of three factors: cue, sex and age on: (1) the observed data: whether the subjects wake up or not, and if they wake up, the mean response time, (2) the model parameters derived from the observed data: (a) the so-called ‘‘matching’’ probability, (b) the wake up threshold, (c) the length of a step in the ‘‘matching’’ process, (3) two derived measures (which are conditional on waking up): the mean number of steps to reach the threshold, and the ninety ninth percentile of the response time. Linear models and, where required, a generalized linear model are used. Analysis of variance (and analysis of deviance for the generalized linear model) are used to determine the statistical significance of the factor effects. Graphical representation of the fitted values provides valuable insights into the factor effects. The relation between the observations, the model parameters and the derived measures is interpreted. Conclusions are drawn and suggestions for further work put forward.
2. Description of the data 2.1. Experiment 1: Responsiveness to auditory and visual cues Participants: Thirty three adult volunteers aged 25–55 years (mean age 43.2 years) who self-reported normal hearing and normal sleep patterns were tested in their homes. There were 15 males and 18 females. Cues: Two auditory cues, a crackling noise akin to the early stage of a timber fire and a ‘shuffling’ noise, as well as a flickering light cue were used. The sounds were
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chosen to represent sounds from a fire. The first is subsequently referred to as ‘‘crackling’’ and the second as ‘‘shuffling’’. The sounds were edited to 30 s from a sound effects recording for movies and videos, with the selection of sections with less variation. The crackling sound varied at the pillow across the 42–48 dBA range with the highest sound being 58 dBA at the 25th second. This sound was sharper and more varied than the shuffling sound, which varied from 43–45 dBA with a few higher points of 52 dBA. The flickering light was directed towards the ceiling and reached the pillow at 5 lux or less. There was considerable variation in the strength of this cue, due to variation in bedroom design and reflectiveness of the ceiling. In some situation the light reading from the pillow was as low as 1 lux. (one lux is approximately the light of one candle flame at a distance of one meter.) Procedure: Cues were presented in random order on three nights within a five night period. Cue delivery occurred five times over a five-minute period lasting for 30 s each time with 30 s between presentations. Participants activated the program (unaware of whether it was delivering a signal that night or not) each night by pressing a ‘sleep’ button on the monitor when they were preparing to go to sleep. Cue presentation began 240 min after this time. This four-hour time period was chosen to increase the likelihood of participants being in Stage 2 or REM sleep when the cues were delivered. These two stages have similar arousal levels [6]. If awakenings were timed to occur earlier in the night then higher arousal thresholds would be expected as deeper sleep would be more likely. (In a normal night’s sleep the deeper Stage 4 sleep is more likely to occur in the first third of sleeping while Stage 2 and REM occur in the last two-thirds.) Participants were told that when they were asleep a cue might be presented that they could see, hear or smell. There would be some nights without cues, though the number could not be stated. Nor could they be told when the signal might occur. If they noticed anything like a signal, they were to press the button by their bed three times. This would stop the delivery and they could go back to sleep. There would be no further cues that night. 2.2. Experiment 2: Responsiveness to an olfactory cue Participants: Twenty young adults, aged 18–26 years (mean age 21.3 years), selfreporting a normal sense of smell and the ability to sleep during the day were involved. There were ten females and ten males. The participants were students from the University. All subjects were asked to restrict their sleep to around 6 h (e.g. from 1 am to 7 am) on the previous night to enhance the likelihood of going to sleep in the afternoon. During preparation for the sleep recordings the olfactory sense of smell of all subjects was somewhat informally checked by determining whether they could smell the alcohol on a cleaning swab. One person could not (due to a cold) and she was eliminated from the study. Cue: The cue presented was the somewhat unpleasant smell of a chemical associated with a smoke taste in food flavourings (Guaiacol). This was prepared in an ethanol base at a 5% concentration level in an aerosol container. A dispenser
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emitted puffs of the mixture at the rate of nine puffs per minute and this odour was dispersed into the bedroom via a duct and fan. Low levels were used, where the parts per million were an average of 0.9 ð0:4Þ at one minute to 6.0 pm ð1:0Þ at 10 min at the pillow. As the human nose is very sensitive, the smell was clearly evident at the pillow 20 s after dispensing began when the concentrations were well below 1 pm. Procedure: Participants attended the sleep laboratory for an ‘afternoon nap’. Sleep stages were recorded via the standard electrode montage on the scalp and face. Subjects began their nap at between 1.30 and 2.15 pm, odour presentations began 90 s after the person entered Stage 2 sleep and continued for a maximum of 10 min. Subjects were asked to press a button three times if they became aware of the presence of a stimulus that they could see, smell or hear. The study provided information about both the time of EEG wakefulness and the behavioural response (button press). Interestingly, three subjects clearly entered EEG wakefulness during the smell presentation but did not press the button. Questioning revealed that they did not believe they had woken up. Accordingly, since this study is essentially behavioural, the three subjects were classified as not waking up.
3. Analysis background The analysis presented in this paper concentrates on the response to single cues. Accordingly, the responses of the 33 subjects in experiment 1 to the three different cues (auditory and visual) were analyzed as if they had been responses from 99 different subjects. For each subject, the age and sex was recorded. The number of subjects responding to all the cues (including the olfactory cue of Experiment 2), after gaps in the data were removed, is shown in Table 1. The total number was 104.
4. Statistical tools for analysis of waking probability To analyse the waking probability it is necessary to investigate the dependence of a variable that takes just two values: waking and not waking, on a continuous variable: age, as well as on various levels of two categorical factors: sex and cue. The
Table 1 Number of subjects by sex and cue Cue
Male
Female
Light Crackle Shuffle Smell
13 12 12 10
15 17 15 10
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two values of the response are coded as 0 (not waking) and 1 (waking). This type of dependent variable is known as a binomial variable. To analyse such a set of data, the classical linear model is not suitable, because it presupposes that the dependent variable is normally distributed for each combination of levels and that its variance is constant. The appropriate model to carry out such an investigation is known as the generalized linear model. The required theoretical foundation was developed by Nelder and Wederburn [7]. For a full exposition of the generalized linear model see [8]. The difference between the linear model and the generalized linear model is as follows: The linear model provides a way of estimating a response variable,Y, conditional on a linear function of the values, x1 ; x2 ; . . . ; xp , of some set of predictors variables. The variance of Y is assumed constant. The generalized linear model provides a way to estimate a function, gð Þ, (called the link function) of the mean response m as a linear function of the values of some set of predictors. The variance of Y is a function V ð Þ of the mean response m. In this paper, the logistic model, for which the link function (the so-called logit function) is given by gðmÞ ¼ logðm=ð1 mÞÞ and the variance function is V ðmÞ ¼ mð1 mÞ. It is used to deal with a binomial response. In the generalized linear model, the sum of squares attributed to each underlying factor is replaced by a deviance term that has a similar interpretation. It is however important to note that each one of the successive deviance terms represents the additional variability that is attributable to the considered factor, once the variability attributable to the previous factors has been accounted for. On account of this, the size of the various deviance components depends somewhat on the order of the terms in the regression equation, which must be chosen judiciously. The significance of each factor is tested by means of a chi squared test. The generalized linear model allows for the study of the interaction between factors. However, because of the comparatively small amount of data, no interactions were included. The computer program used to implement the analyses was S-PLUS Version 6.
5. Results for the waking probabilities The binomial response variable was regressed on two categorical factors: cue and sex, and on age as a continuous variable. The regression on age was carried out as a polynomial regression. After some experimentation it turned out that a polynomial of order three would be suitable. The most significant factor was the cue and the second sex. The analysis of deviance is shown in Table 2. In the rest of the paper, the estimated waking probability will be denoted by pZ . It can be seen that while the cue and sex factors are significant at the 2% level, age is not significant, even at the standard 5% level, although there is clearly some effect. The effect of cue, sex and age are illustrated in Figs. 1 and 2 where the fitted values of the response probability are plotted against cue and age for males and females,
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Table 2 Analysis of deviance of waking probability
NULL Cue Sex Age
Degrees of freedom
Deviance
Residual deviance
3 1 3
20.50 5.66 4.34
134.17 113.66 108.00 103.66
Probability of chi squared
0.0001 0.017 0.23
WAKING PROBABILITIES FOR MALES BY AGE AND CUE 1.00
CRACKLE
Probability of waking
0.80
0.60 SHUFFLE
0.40
0.20
SMELL
LIGHT
0.00 20
30
40
50
Age Fig. 1. Fitted waking probabilities of male subjects by age and cue.
respectively. Continuous curves are obtained by using cubic spline interpolation to derive intermediate values between the observations. A larger sample would have provided more reliable results for the effect of age. With the data at hand, any conclusions about the effect of age should be taken as preliminary only. In order to tabulate the waking probabilities by cue, sex and age, subjects were grouped by age as shown in Table 3. Table 4 gives the fitted values of the waking probabilities classified by age group, sex and cue. The table shows that the mean female waking probability is consistently larger than the mean male waking probability for each cue and each age group,
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1.00
669
WAKING PROBABILITIES FOR FEMALES BY AGE AND CUE CRACKLE
0.80 Probability of waking
SHUFFLE
0.60 SMELL
0.40
LIGHT
0.20
0.00 20
30
40
50
Age Fig. 2. Fitted waking probabilities of female subjects by age and cue.
Table 3 Age groups and number of subjects in each age group Age group
Number
p30 31–42 43–46 47–50 51þ
30 23 19 21 11
Table 4 Fitted waking probabilities by age group, sex and cue Cue
Light Crackle Shuffle Smell
Sex
Male Female Male Female Male Female Male Female
Age Group p30
31–42
43–46
47–50
51þ
All ages
0.267 0.522 0.848 0.944 0.690 0.870 0.317 0.583
0.170 0.381 0.758 0.904 0.556 0.790 — —
0.224 0.464 0.816 0.930 0.638 0.841 — —
0.356 0.624 0.894 0.962 0.772 0.910 — —
0.546 0.783 0.949 0.982 0.880 0.957 — —
0.288 0.551 0.836 0.939 0.673 0.862 0.317 0.583
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Table 5 Analysis of variance of response time
Cue Sex Age Residuals
Degrees of freedom
Sum of squares
Mean squares
F value
Pr(F)
3 1 3 60
117,793 2,659 22,850 177,639
39,264 2,659 7,617 2,961
13.3 0.90 2.57
0.000001 0.35 0.06
6. Results for the mean response time Once the subjects who did not wake up were eliminated, there remained a sample of 68 subjects for whom a response time (in seconds) had been recorded. The analysis was carried out using a standard linear model: the response time was regressed as before on two categorical factors: cue and sex, and on age as a continuous variable (represented as in the previous section by a polynomial of order 3). The analysis of variance is given in Table 5. Here the conventional sum of squares analysis is carried out, and significance is tested by means of an F test. It can be seen that the type of cue remains the most significant factor. However, it is now age that takes the second place and just misses becoming significant, while sex is no longer significant. The effect of cue and age is illustrated in Fig. 3 where the fitted values of the mean response time are plotted against cue and age. Continuous curves are obtained by using cubic spline interpolation to derive intermediate values between the observations. A larger sample would have provided more reliable results for the effect of age. With the data at hand, any conclusions about the effect of age and sex should be taken as preliminary only. Table 6 gives the fitted values of the mean response time classified by age group, sex and cue, using the same age groups as for the waking probabilities. The table shows that the mean female response time is consistently shorter than the mean male response time for each cue and each age group, even though here this difference does not satisfy the criteria of statistical significance.
7. The stochastic model for the response and its parameters One aim of the research reported in this paper is to examine the dependence of the parameters of the stochastic model for the response to fire alarms proposed in [1] on the three considered factors: cue, sex and age. This will provide valuable insight into the underlying determinants of the response variability. As mentioned in Introduction, the model used in this paper is called a memory retrieval theory because it considers an item recognition task as testing a single probe item against a group of items in memory which has been designated as the memory
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COMBINED MEAN RESPONSE TIME BY AGE AND CUE SMELL
Mean response time (secs)
150
LIGHT 100
50
CRACKLE
SHUFFLE 0 20
30
40
50
Age Fig. 3. Combined fitted mean response time by age and cue.
Table 6 Fitted mean response time by age group, sex and cue Cue
Light Crackle Shuffle Smell
Sex
Male Female Male Female Male Female Male Female
Age Group p30
31–42
43–46
47–50
51þ
All ages
— 116.86 71.77 57.98 — 52.28 150.52 136.74
93.26 — 34.38 20.59 28.68 14.89 — —
84.58 70.79 25.70 11.91 20.00 6.21 — —
85.57 71.78 26.69 12.90 20.99 7.20 — —
119.38 105.59 60.50 46.71 54.80 41.01 — —
98.57 85.59 38.14 25.16 32.04 19.05 150.52 136.74
search set. The retrieval process consists of comparing a collection of features of each element of the memory search set with the probe item. In the application of the model to the response of subjects to a fire cue the probe item is identified with the stimulus induced in the sleeping subject by the cue. This stimulus is compared with the items of the memory search set. Comparison of a probe to a memory-set item proceeds by the gradual accumulation of evidence, that is, information representing the goodness of match, over time. It is easiest to conceptualize the comparison process as a feature-matching
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process in which probe and memory-set item features are matched one by one. A count is kept of the combined sum of the number of feature matches and nonmatches, so that for a feature match, a counter is incremented, and for a feature nonmatch, the counter is decremented. The counter begins at some starting value Z, and if a total of A counts are reached, the probe is declared to match the memory-set item (A Z more feature matches than nonmatches). But if a total of zero counts are reached, an item nonmatch is declared. The response is thus modelled by a random walk bounded from above and below. The lower boundary is taken to be 0. The upper boundary is denoted by A and the starting value by Z. The process ends when either of the boundaries is reached. If the lower boundary is reached first the subject dismisses the stimulus and continues to sleep. If, on the other hand, the upper boundary is reached first the subject wakes up. Thus, the upper boundary A can be thought of as a threshold of sensitivity to the stimulus. The random walk is described by four parameters: (1) Z, the starting value, (2) A, the wake up threshold, (3) the probability p of a match between the stimulus and an item of the memory set, resulting in an upward jump of one unit at each step. A non-match, that occurs with probability 1 p, results in a downward jump of one unit. (4) the length d of one step. It was decided, after some preliminary analysis, to fix the starting value Z to an appropriate value. The three remaining parameters can then be estimated from the knowledge of: (1) the probability pZ of the subjects responding to the alarm, (2) the mean and standard deviation of the response time, for subjects who have responded.
8. The variance of the response time So far in the paper, the probability of response and the mean response time have been determined for each category of subjects. But because of the comparatively small sample available, it is not possible to estimate the standard deviation of the response time separately for each category with any confidence. However, in view of the fact that the cue factor is highly significant for the response time, the reasonable path is to estimate the standard deviation for each cue, using the sample of all subjects receiving the cue. It is estimated as the standard deviation of the difference between the observed response time and the fitted mean. The estimated standard deviations are given in Table 7. The overall standard deviation is 51.49 s.
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Table 7 Estimated standard deviation for each cue Cue
Standard deviation
Light Crackle Shuffle Smell
92.71 28.23 22.86 81.18
9. Results for A, p and d The value of Z was chosen to be 50. This was a compromise between two conflicting considerations. On the one hand, choosing Z too small would make the random walk too coarse. On the other hand, making Z too large would vastly increase the amount of computation required without achieving a more significant fit. The choice was also in line with the results of the analysis in [1], where A was chosen to be 100 and the two values of Z evaluated were 75 and 72. Using the fitted response probability, the fitted mean, and the estimated standard deviation, values of A, p and d (in seconds) were calculated for each subject who woke up and each cue, using the method described in [1]. Dependence of A, p and d on cue, sex and age was analysed, using a standard linear model, with cue and sex being treated as categorical factors and age as a continuous variable, modelled as previously with a polynomial of order three. It would appear at first sight that the analysis on p should be carried out using the logistic model, as p is a probability. However, closer analysis of p reveals that its values are clustered around the value 0.5. Indeed, 57 out of 68 values of p lie in the interval (0.45,0.55). The histogram of these 57 values is given by in Fig. 4. But it is well-known that the logistic transformation is practically linear when the dependent variable is in the neighbourhood of 0.5, so that using it does not improve the fit over the standard linear model. On the other hand, the standard linear model is more robust than the generalized linear model because it makes fewer distributional assumptions. It was therefore decided to use the standard linear model for p as well. The analysis of variance for A, d and p is given in Tables 8–10. The analysis of variance for the threshold A shows that age is by far the most significant factor, followed by cue. Sex just misses out to be significant at the 5% level. The effect of cue, sex and age on A are illustrated in Figs. 5 and 6. Since the effect of sex turned out to be barely significant, the effect of cue and age on A for the combined sample of males and females is also illustrated in Fig. 7. As in the previous figures, continuous curves are obtained by using cubic spline interpolation. Table 11 gives the fitted values of A classified by age group, sex and cue. The table shows that the mean female A is consistently lower than the mean male A for each cue and each age group. A larger sample would probably have yielded a significant difference.
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HISTOGRAM OF p FOR 0.45
25
20
15
10
5
0 0.45
0.47
0.49
0.51
0.53
0.55
p
Fig. 4. Histogram of p for 0:45opo0:55.
Table 8 Analysis of variance of A
Cue Sex Age Residuals
Degrees of freedom
Sum of squares
Mean squares
F value
Pr(F)
3 1 3 60
35,337 4,629 54,778 71,056
11,779 4629 18,259 1,184
9.95 3.9 15.4
0.00002 0.053 0.00
Table 9 Analysis of variance of d
Cue Sex Age Residuals
Degrees of freedom
Sum of squares
Mean squares
F value
Pr(F)
3 1 3 60
130.1 0.116 47.46 738.4
43.4 0.116 15.82 12.31
3.52 0.009 1.28
0.020 0.92 0.29
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Table 10 Analysis of variance of p
Cue Sex Age Residuals
Degrees of freedom
Sum of squares
Mean squares
F value
Pr(F)
3 1 3 60
0.069 0.0028 0.0083 0.40
0.023 0.0028 0.0028 0.0067
3.48 0.43 0.42
0.02 0.52 0.74
FITTED A FOR MALES BY AGE AND CUE CRACKLE
150 SMELL
Fitted.A
SHUFFLE
100 LIGHT
50
0 20
30
40
50
Age Fig. 5. Fitted A for male subjects.
The main conclusions from the analysis of the dependence of the threshold A on cue and age are as follows: (1) The order of the cues in descending values of fitted A is shuffle, crackle, light and smell. (2) The lowest values of A for all cues (except for smell, for which the range of ages does not extend beyond 26) are in the early forties, with A increasing with younger as well as with older age. It is possible that the higher threshold for younger people can be attributed to an increased delta power (see [9]). This means that their sleep is deeper and consequently a larger net number of matches is required for response. On the other hand, the higher threshold for older people
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FITTED A FOR FEMALES BY AGE AND CUE
150
Fitted.A
SHUFFLE 100
SMELL CRACKLE
LIGHT
50
0 20
30
40
50
Age Fig. 6. Fitted A for female subjects.
COMBINED A BY AGE AND CUE
150
Fitted.A
SMELL
CRACKLE
SHUFFLE
100
LIGHT
50
0
20
30
40 Age
Fig. 7. Combined fitted A.
50
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Table 11 Values of fitted A by age group, sex and cue Cue
Sex
Light Crackle Shuffle Smell
Age Group
Male Female Male Female Male Female Male Female
p30
31–42
43–46
47–50
51þ
— 132.74 179.83 157.92 — 174.43 166.50 144.59
75.46 — 100.63 78.72 117.15 95.24 — —
58.95 37.04 84.13 62.22 100.64 78.73 — —
65.79 43.88 90.96 69.05 107.48 85.57 — —
111.76 89.85 136.94 115.03 153.45 131.54 — —
Table 12 Mean values of p
p Meanj0:5 pj
Light
Crackle
Shuffle
Smell
0.46 0.0417
0.54 0.0430
0.48 0.05
0.50 0
Table 13 Mean values of d
Mean d
Light
Crackle
Shuffle
Smell
0.21
0.14
0.06
0.030
might be attributed to a requirement for stronger evidence before they decide to respond. In the analysis of variance of p and d, it turned out that only cue was significant, while sex and age were not. Two values of d, 26.02 and 16.50,were considered to be outliers. Their appearance is probably due to the very rough estimation of the standard deviation as well as the fact that the values of A, p and d were calculated from a discrete approximation that could be misleading for certain values of the input. After their removal the 66 remaining values of d lay in the interval (0.007,1.18). The mean values of p and d for each cue are shown in Tables 12 and 13. The order of the cues in descending values of mean p is: crackle, smell, shuffle, light.
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COMBINED FITTED p BY AGE AND CUE
CRACKLE
Fitted.p
0.55
0.50
SMELL SHUFFLE
0.45
LIGHT
0.40 20
30
40
50
Age Fig. 8. Combined fitted p.
Although the dependence of p on age is not statistically significant, it is nevertheless of interest to look at the graph of fitted p versus cue and age. It is given by Fig. 8. There is also some difference between the mean values of p for males and females. They are 0.494 and 0.506, respectively. The order of cues in descending values of mean d is: light, crackle, shuffle, smell. What is particularly noticeable, (and perhaps somewhat puzzling) is the inordinately high value of d for light, indicating that, in the context of waking up to a light cue, a visual stimulus is processed far more slowly than an auditory or olfactory stimulus.
10. Results for the number of steps to response While the probability of response depends only on the two underlying parameters A and p, the response time depends on all three parameters A, p and d. This makes it difficult to interpret the dependence of the response time on cue, sex and age in terms of the dependence of the underlying parameters. One way of simplifying the interpretation is to evaluate the number of steps to response, which will be denoted by N. This is easily accomplished by dividing the response time for each subject by the evaluated length of step d. Analysis of the dependence of N on cue, sex and age was carried out, using a standard linear model with cue and sex treated as categorical factors and age as a
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Table 14 Analysis of variance of N
Cue Sex Age Residuals
Degrees of freedom
Sum of squares
Mean squares
F value
Pr(F)
3 1 3 60
225,605,699 37,585,271 62,758,913 290,354,873
75,201,900 37,585,271 20,919,638 4,839,248
15.5 7.8 4.3
0.0000001 0.007 0.008
FITTED N FOR MALES BY AGE AND CUE SMELL
Fitted.N
6,000
4,000
SHUFFLE CRACKLE
2,000
LIGHT
0
20
30
40
50
Age Fig. 9. Fitted N for male subjects.
continuous variable modelled as previously with a polynomial of order three. The analysis of variance is given in Table 14. Here it turns out that all three factors are highly significant. The effect of cue, sex and age on N is illustrated in Figs. 9 and 10.
11. The ninety ninth percentile of the distribution of response time It was pointed out in [1] that one of the most important aims of the stochastic analysis of the response time to fire cues was to estimate the high quantiles of the distribution of the response time, since it is those who respond the most slowly who are the most at risk.
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FITTED N FOR FEMALES BY AGE AND CUE
SMELL
Fitted.N
6000.00
4000.00
2000.00 SHUFFLE CRACKLE 0.00
LIGHT 20
30
40 Age
50
Fig. 10. Fitted N for female subjects.
Following [1], the ninety ninth percentile of the distribution, denoted by q99 , was taken as a representative of the upper tail of the distribution. It was evaluated for all subjects who woke up, using the values of A, p and d previously obtained and the formulae in [1]. There were eight subjects where the value of p was so low that the value of q99 was essentially infinite. They were excluded from the sample, leaving 60 subjects for analysis. In a risk analysis of response, those eight subjects should be classified as not waking up. Analysis of the dependence of q99 on cue, sex and age was carried out, using a standard linear model with cue and sex treated as categorical factors and age as a continuous variable modelled as previously with a polynomial of order three. The analysis of variance is given in Table 15. Here it turns out that while cue and sex are significant, age is not. This is consistent with the results of Table 2, which gives the analysis of deviance of the waking probability, since the subjects who do not wake up may be thought of as subjects with an infinite response time. The effect of cue, sex and age on q99 is illustrated in Figs. 11 and 12. Table 16 gives the fitted values of q99 classified by age group, sex and cue. The table shows that the mean female q99 is consistently lower than the mean male q99 for each cue and each age group.
12. Summary of dependence on cue, sex and age The dependence of the observed variables pZ and T as well as the model parameters A, p and d, the derived variable N and the ninety ninth percentile q99 on the three factors considered, cue, sex and age, is summarized in Table 17.
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Table 15 Analysis of variance of q99
Cue Sex Age Residuals
Degrees of freedom
Sum of squares
Mean squares
F value
Pr(F)
3 1 3 52
1726390 12090 12092 104349
575464 12090 4031 2006.7
287 6.02 2.01
0.00000 0.017 0.124
99TH QUANTILE OF RESPONSE TIME FOR MALES
600
99th quantile (secs)
LIGHT
SMELL 400
CRACKLE
200
SHUFFLE 0 20
30
40
50
Age Fig. 11. Fitted q99 for male subjects.
13. Theoretical dependence of pZ and N on p and A To be able to interpret the relationship between the observed parameters, namely the probability of waking and the mean response time, in terms of the model parameters A, p and d, it is necessary to look at the theoretical relationship between them. Fig. 13 illustrates the dependence of the probability of waking pZ on p and A while Fig. 14 illustrates the dependence of the mean number of steps N on p and A. From Fig. 13 we see that pZ varies directly with p and inversely with A. This is in accordance with common sense. However, there is much difference between the sensitivity of pZ to A and to p. This is illustrated in Figs. 15 and 16, which plot the values of the derivatives of logðpZ Þ with respect to logðAÞ and logðpÞ against p.
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99TH QUANTILE OF RESPONSE TIME FOR FEMALES
99th quantile (secs)
600
SMELL
LIGHT
400
200
CRACKLE
SHUFFLE
0 20
30
40
50
Age Fig. 12. Fitted q99 for female subjects.
Table 16 Values of fitted q99 by age group, sex and cue Cue
Light Crackle Shuffle Smell
Sex
Male Female Male Female Male Female Male Female
Age Group p30
31–42
43–46
47–50
51þ
All ages
— 491.23 182.15 143.01 — 116.23 506.83 467.69
480.27 — 132.04 92.90 105.27 66.12 — —
— 446.27 137.19 — 110.42 71.27 — —
511.43 472.29 163.21 124.07 136.43 97.29 — —
503.28 464.14 155.06 115.91 128.28 89.14 — 467.69
500.91 471.71 147.07 117.88 117.43 88.24 506.83
It must be pointed out that these derivatives measure the relative change of pZ with respect to the relative change of A and p, which allows comparison of sensitivity with respect to the variables by direct comparison of the derivatives. It can be seen that pZ is far less sensitive to relative variations in A than it is to relative variations in p, particularly for p near 0.5. On the other hand, from Fig. 14 we see that while N varies directly with A, it varies inversely with j0:5 pj, i.e., with the distance of p from 0.5. Superficially, one would
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Table 17 Summary of dependence on cue, sex and age Dependence on
Waking probability pZ Mean response time T Threshold A Probability of match p Length of step d Mean number of steps to waking N Ninety ninth percentile q99
Cue
Sex
Age
Strong Strong Strong Strong Strong Strong Strong
Significant Weak Borderline significant Very weak Very weak Strong Strong
Weak Borderline significant Strong Very weak Weak Strong Weak
DEPENDENCE OF pZ ON p AND A 1.0
0.8
pZ
0.6
0.4 A=75
0.2
A=100 A=150
0.0 0.46
0.47
0.48
0.49 p
0.50
0.51
0.52
Fig. 13. Dependence of pZ on p and A.
have thought that N should vary inversely with p monotonically. This, however, ignores the fact that N is the mean response time conditional on waking up. The average is taken only over the sample paths that end up in waking up. For small values of p most subjects do not wake up. The only subjects who wake up are those who at the beginning of the matching process happen to have a large enough proportion of successful matches at the beginning of the process to bring them to the threshold A. This tends to shorten the response time conditional on waking up.
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DEPENDENCE OF N ON p AND A A=150
6000 A=125
N
4000 A=100
2000 A=75
0 0.43
0.45
0.47
0.49
0.51
0.53
0.55
0.57
p Fig. 14. Dependence of N on p and A.
RELATIVE DERIVATIVE OF pZ W.R.T. A
d(log pZ)/d(log A)
0
-10 A=75
-20 A=100
-30 A=150
-40 0.43
0.45
0.47
0.49
0.51
0.53
0.55
p Fig. 15. Dependence of the derivative of logðpZ Þ w.r.t. logðAÞ on p.
0.57
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RELATIVE DERIVATIVE OF pZ W.R.T. p 200 A=150
d(log pZ)/d(log p)
150
100 A=100 50 A=75 0 0.43
0.45
0.47
0.49
0.51
0.53
0.55
0.57
p Fig. 16. Dependence of the derivative of logðpZ Þ w.r.t. logðpÞ on p.
14. Interpretation of the variation of waking probability in terms of A and p As pointed out in Section 13, the waking probability depends only weakly on the threshold A, compared to its dependence on p. Indeed Figs. 1 and 2 look remarkably like Fig. 8 as far as order of cues and age dependence is concerned. So we can conclude that the dependence on cue and on age is governed by the matching probability p. Similarly, the difference in waking probabilities between males and females can be attributed, at least in part, to the slightly higher value of p for females, as compared to males. Here, however, the lower threshold for females compared to males will tend to reinforce the effect of a higher value of p. The dependence of pZ on cue, sex and age in terms of A and p is summarized in Table 18.
15. Interpretation of the variation of mean number of steps to response, N, in terms of A and p Here the interpretation is rather straightforward in the light of the comments in Section 13. The general shape of the dependence of N on cue, sex and age is similar to the shape of the dependence of the threshold A. On the other hand, the larger values of meanj0:5 pj for shuffle and crackle compared to light will tend to shift the
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Table 18 Summary of dependence of pZ on cue, sex and age Dependence of pZ in terms of
Cue Sex Age
A
p
No Yes No
Yes Yes Yes
Table 19 Summary of dependence of N on cue, sex and age Dependence of N in terms of
Cue Sex Age
A
p
Yes Yes Yes
Yes No No
curves downwards, and the small value of meanj0:5 pj for smell tends to shift it higher. The dependence of N on cue, sex and age in terms of A and p is summarized in Table 19.
16. Conclusions The main use of the analysis in this paper in fire engineering is to provide precise data on two quantities: (1) the probability of not waking up, (2) the ninety ninth percentile of the response time for the subjects who have woken up. The latter quantity is particularly important as it determines to what extent fire retardation measures should be taken to allow the subjects who wake up to evacuate the fire area before untenable conditions set in. However, that quantity cannot be evaluated without first evaluating the model parameters. In addition, knowledge of the model parameters is needed to evaluate the full probability distribution of the response time, which is needed if the response time study is extended to multiple cues, as will be proposed in the next section. The analysis in the preceding sections indicates a very significant difference between males and females, both in waking probability and in the response time. Females have a waking up probability consistently higher than that of males in the same categories, which is stastistically significant. Among those who wake up,
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the mean response time of females is consistently shorter than that of males, although the difference is not statistically significant. When it comes to the ninety ninth percentile, however, the response time of females is again consistently shorter than that of males, and the difference is statistically highly significant. In terms of the model parameters, it appears that the greater sensitivity of females is mainly due to a higher value of the matching probability p, although the analysis fails to attribute a statistical significance to the difference. As far as the response time is concerned, the difference is reinforced by the consistently lower value of the threshold for females. The variable of age exerts little influence on the waking probability and an influence on the mean response time of borderline significance. The latter appears to be mainly due to the dependence of the threshold A on age. That threshold is high for the young and the old, reaching a minimum in the early forties. A possible explanation, already given in Section 9, is that it is possible that the higher threshold for younger people can be attributed to an increased delta power. This means that their sleep is deeper and consequently a larger net number of matches is required for response. It is speculated that the higher threshold for older people might be attributed to a requirement for stronger evidence before they decide to respond. The difference between age groups is far less for the ninety ninth percentile of the response time. The most significant difference in the analysis is between the four cues investigated. Indeed, observations as well as model parameters and derived values all show clear differences. It must be pointed out, however, that response to a cue clearly depends on the intensity of the cue. Since the intensity of cue was not varied in the experiment being analysed, it must be kept in mind that all comparisons between cues apply only to the intensities used in the experiment and that these were low level intensities. As far as waking probabilities are concerned, the most effective cues were auditory (crackle, followed by shuffle). The two least effective cues were light and smell. Similarly, for the shortest mean response times as well as the shortest ninety ninth percentile of response time, the cues were again auditory (crackle and shuffle) with smell and light requiring a longer response time. The difference in waking probabilities is clearly attributable to the difference in matching probabilities. However, analysis shows that the threshold for light is actually lower than that of the auditory cues, and the value of meanj0:5 pj is similar for light and the auditory cues. Nevertheless, the response time for light is longer. It turns out that this is due to a much larger value of the step length d for the visual cue. This might be due to the fact that the amount of information contained in visual cues is very different from that contained in auditory or olfactory cues, requiring more time for information processing of the cue. The practical conclusion of the study is that (1) globally speaking the waking probability is unacceptably low for visual and olfactory cues taken singly at the low intensities of this study, making the risk of death unacceptably high,
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(2) from a fire engineering point of view the existence of low level flickering light and fire smells is very unlikely to arouse sleeping people.
17. Proposals for further study The authors propose to conduct in the near future further experiments to evaluate (1) the effect of multiple cues, (2) the effect of varying the intensity of cues, (3) the effect of alcohol. Extensive further experimentation in the area of the processes involved in waking up, coupled with detailed modelling and analysis, is the only way to acquire the knowledge required to develop performance-based guidelines for the design of fire safe dwellings. References [1] Hasofer AM. A stochastic model for the time to awaken in response to a fire alarm. J Fire Protect Eng 2001;11(3):151–60. [2] Ratcliff R, Murdock Jr BB. Retrieval processes in recognition memory. Psychol Rev 1972;83:190–214. [3] Bruck D, Bliss RA. Sleeping children and smoke alarms. Fourth asian-oceania symposium on fire science and technology. Tokyo, May 2000. [4] Bruck D, Horasan M. Non-arousal and non-action of normal sleepers in response to a smoke detector alarm. Fire Safety J 1995;25:125–39. [5] Bruck D, Brennan P. Recognition of fire cues during sleep. In: Shields J, editor. Proceedings of the second international symposium on human behaviour in fire. London: Interscience Communications; 2001. p. 241–52. [6] Zepelin H, McDonald CS, Zammit GK. Effects of age on auditory awakening. J Gerontol 1984;39(3):294–300. [7] Nelder JA, Wedderburn RWM. Generalized linear models. J Roy Stat Soc Ser A 1972;135:370–84. [8] McCullagh P, Nelder JA. Generalized linear models, 2nd ed. London: Chapman and Hall; 1989. [9] A˚stro¨m C, Trojaborg W. Relationship of age to power spectrum analysis of EEG during sleep. J Clin Neurophysiol 1992;9:424–30.